On Computing the Number of Latin Rectangles

Doyle (circa 1980) found a formula for the number of $$k \times n$$k×n Latin rectangles $$L_{k,n}$$Lk,n. This formula remained dormant until it was recently used for counting $$k \times n$$k×n Latin rectangles, where $$k \in \{4,5,6\}$$k∈{4,5,6}. We give a formal proof of Doyle’s formula for arbitrary k. We also improve a previous implementation of this formula, which we use to find $$L_{k,n}$$Lk,n when $$k=4$$k=4 and $$n \le 150$$n≤150, when $$k=5$$k=5 and $$n \le 40$$n≤40 and when $$k=6$$k=6 and $$n \le 15$$n≤15. Motivated by computational data for $$3 \le k \le 6$$3≤k≤6, some research problems and conjectures about the divisors of $$L_{k,n}$$Lk,n are presented.